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Convex Bodies: The Brunn–Minkowski Theory

2nd Edition


Part of Encyclopedia of Mathematics and its Applications

  • Date Published: October 2013
  • availability: Available
  • format: Hardback
  • isbn: 9781107601017

£ 133.00

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About the Authors
  • At the heart of this monograph is the Brunn–Minkowski theory, which can be used to great effect in studying such ideas as volume and surface area and their generalizations. In particular, the notions of mixed volume and mixed area measure arise naturally and the fundamental inequalities that are satisfied by mixed volumes are considered here in detail. The author presents a comprehensive introduction to convex bodies, including full proofs for some deeper theorems. The book provides hints and pointers to connections with other fields and an exhaustive reference list. This second edition has been considerably expanded to reflect the rapid developments of the past two decades. It includes new chapters on valuations on convex bodies, on extensions like the Lp Brunn–Minkowski theory, and on affine constructions and inequalities. There are also many supplements and updates to the original chapters, and a substantial expansion of chapter notes and references.

    • Extensively revised to reflect developments in the field since the publication of the first edition in 1993
    • Comprehensive coverage serves newcomers in the field as well as experienced researchers
    • Introductory material has been tried and tested for classroom use
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    Reviews & endorsements

    Review of the first edition: 'Neither one of [the old classics] may be considered a substitute for the excellent detailed monograph written by Rolf Schneider. I recommend this book to everyone who appreciates the beauty of convexity theory or who uses the strength of geometric inequalities, and to any expert who needs a reliable reference book for his/her research.' V. Milman, Bulletin of the American Mathematical Society

    Review of the first edition: 'Professor Schneider's book is the first comprehensive account of the Brunn-Minkowski theory and will immediately become the standard reference for the Aleksandrov-Fenchel inequalities and the current knowledge concerning the cases of equality and estimates of their stability. The book is aimed at a broad audience from graduate students to working professionals. The presentation is very clear and I enjoyed reading it.' Bulletin of the London Mathematical Society

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    Product details

    • Edition: 2nd Edition
    • Date Published: October 2013
    • format: Hardback
    • isbn: 9781107601017
    • length: 760 pages
    • dimensions: 236 x 163 x 46 mm
    • weight: 1.29kg
    • availability: Available
  • Table of Contents

    Preface to the second edition
    Preface to the first edition
    General hints to the literature
    Conventions and notation
    1. Basic convexity
    2. Boundary structure
    3. Minkowski addition
    4. Support measures and intrinsic volumes
    5. Mixed volumes and related concepts
    6. Valuations on convex bodies
    7. Inequalities for mixed volumes
    8. Determination by area measures and curvatures
    9. Extensions and analogues of the Brunn–Minkowski theory
    10. Affine constructions and inequalities
    Appendix. Spherical harmonics
    Notation index
    Author index
    Subject index.

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    Convex Bodies: The Brunn–Minkowski Theory

    Rolf Schneider

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  • Author

    Rolf Schneider, Albert-Ludwigs-Universität Freiburg, Germany
    Rolf Schneider is Professor Emeritus in the Faculty of Mathematics and Physics at Albert-Ludwigs-Universität Freiburg, Germany.

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